# Regularity properties of the minimum-time map

• Gianna Stefani
Chapter
Part of the Progress in Systems and Control Theory book series (PSCT, volume 9)

## Abstract

The aim of this paper is to give a survey on some known results concerning the regularity properties of the minimum-time map around an equilibrium point of a control system and to discuss the links of these properties with the viscosity solutions of the Hamilton Jacobi Gellman equation. For sake of simplicity let us consider a control system on R n defined by:
$$(\sum )\dot X = f(X,u) \equiv {f_0}(X) + \sum\limits_{i = 1}^m {{u_i}} {f_i}(X),X(0) = {X_0}$$
where the fi’s are C vector fields and the control map u = (u1,...,um) belongs to the class u of the integrable maps with values in the set
$$\Omega = \left\{ {({\omega _1}, \ldots ,{\omega _m}) \in {R^m}:\left| {{\omega _i}} \right| \leqslant 1,i = 1, \cdots ,m} \right\}.$$

## Keywords

Local Controllability Viscosity Solution Regularity Property Nonlinear Control System Grade Vector Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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